Forgive the recent apparent obsession (I’d call it a fascination) with the #cycloid but I’ve just discovered something I’d not heard of before. It is also called a #TautochroneCurve or #Isochrone curve, which means that a particle starting from any location on the curve will get to the #MinimumPoint at precisely the same time as a particle starting at any other point.
For a long time I felt like I didn't really understand the Yoneda Lemma. I knew some things that people said about it ('we can understand objects by the maps into them' and 'the Yoneda embedding is full and faithful') but the statement 'Hom(Hom(A, -), F) = F(A)' itself was something I could only use as a symbolic manipulation without understanding.
On the other hand, I did separately know facts like 'In the category of quivers there are objects which look like • and •→•, such that the maps out of them tell you exactly the vertices and edges in your quiver' and 'In the category of simplicial sets there are objects which are just an n-simplex; maps out of them are the n-simplices of the object you are mapping into'.
Somehow I only recently realised that these examples are precisely the Yoneda Lemma. These objects are precisely presheaves of the form Hom(A, -), and the Yoneda Lemma tells you what you get when you map out of them.
In particular I think it would be useful to give the quiver example to students when they learn the Yoneda Lemma.
Serbian mathematician, astronomer, climatologist Milutin Milanković was born #OTD in 1879.
He is best known for his theory of climate change (Milankovitch cycles), which explains the long-term cycles in Earth's climate based on changes in its orbit and orientation relative to the Sun. He used his expertise in mathematics to develop detailed models of how these orbital changes influence the distribution of solar radiation on Earth’s surface.
I genuinely think that a fanatical emphasis on real world uses has done more to harm mathematical literacy than any other academic policy.
Kids love weird shit, and teachers should 100% take advantage of that and roll into class bein' like "Ok nerds, here's the math that proves you're a donut"
Here I tried to prove the Existence Theorem for Laplace Transforms. I don't know what the/a "conventional proof" looks like, but this is what I came up with.
French mathematician Abraham de Moivre was born #OTD in 1667.
He is best known for de Moivre's theorem, which links complex numbers and trigonometry, and for his work in the development of analytic geometry and the theory of equations. He published "The Doctrine of Chances" (1718) where he developed a formula for the normal approximation to the binomial distribution, now known as the de Moivre-Laplace theorem.
A couple of weeks ago, I posted an #animation of a point on a circle generating a #cycloid.
If you turn the curve "upside down", you get the #BrachistochroneCurve. This curve provides the shortest travel time starting from one cusp to any other point on the curve for a ball rolling under uniform #gravity. It is always faster than the straight-line travel time.
Anyway, the #animation took a bit of thought as it requires a bit of #Mechanics, some #Integration and is made a bit more tricky as the curve is multi-valued and so you need to treat different branches separately. The #AnimatedGif was produce with #WxMaxima.
Here's something I just learned: the lucky numbers of Euler.
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k² − k + n produces a prime number.
Leonhard Euler published the polynomial k² − k + 41 which produces prime numbers for all integer values of k from 1 to 40.
Only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS).
The Heegner numbers 7, 11, 19, 43, 67, 163, yield prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.
h/t John Carlos Baez
(@johncarlosbaez) for pointing this out.
The fascinating Heegner numbers [1] are so named for the amateur mathematician who proved Gauss' conjecture that the numbers {-1, -2, -3, -7, -11, -19, -43, -67,-163} are the only values of -d for which imaginary quadratic fields Q[√-d] are uniquely factorable into factors of the form a + b√-d (for a, b ∈ ℤ) (i.e., the field "splits" [2]). Today it is known that there are only nine Heegner numbers: -1, -2, -3, -7, -11, -19, -43, -67, and -163 [3].
Interestingly, the number 163 turns up in all kinds of surprising places, including the irrational constant e^{π√163} ≈ 262537412640768743.99999999999925... (≈ 2.6253741264×10^{17}), which is known as the Ramanujan Constant [4].
"It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater ; not because the pleasure it gives (although very pure) is comparable [...] to that of music [...]" – Bertrand Russell (1872–1970) #quote#mathematics#art#maths#math
British mathematician, logician, philosopher, & public intellectual Bertrand Russell was born #OTD in 1872.
One of Russell's most significant achievements is the co-authorship of "Principia Mathematica" (1910-1913) with Alfred North Whitehead. His works, such as "The Problems of Philosophy" (1912) & "Our Knowledge of the External World" (1914), explored issues related to knowledge, perception, & the scientific method.
"Physics is mathematical not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover."
An Outline of Philosophy Ch.15 The Nature of our Knowledge of Physics (1927)
"The pursuit of philosophy is founded on the belief that knowledge is good, even if what is known is painful."
“Since the beginning of #Israel’s war on #Gaza, academics in fields including #politics, #sociology, Japanese #literature, public #health, Latin American and Caribbean studies, Middle East and African studies, #mathematics, #education, and more have been fired, suspended, or removed from the classroom for pro-#Palestine, anti-Israel speech.”